Rationalize the denominator and simplify the following expression:

fraction numerator 4 plus square root of 2 over denominator 2 space minus square root of 2 end fraction


5 plus 3 square root of 2


10 plus 6 square root of 2


2 plus 6 square root of 2


2 plus 3 square root of 2

The unknown angle in the intersecting secant is as follows:

m∠GPE = 27 degrees

Secant is a line that intersects a curve at a minimum of two distinct points.

Using angle of intersecting secant theorem, the angle made by two secants (a line that cuts a circle at two points) that intersect outside the circle is half of the furthest arc minus the nearest arc.

Therefore, m∠GPE can be found as follows;

Hence,

m∠GPE = 1 / 2 (mGE - mHF)

mGE = 97 degrees

mHF = 43 degrees

Therefore,

m∠GPE = 1 / 2 (97 - 43)

m∠GPE = 1 / 2 (54)

m∠GPE =54 / 2

m∠GPE = 27 degrees

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a) X is a binomial random variable because there is a fixed number of trials, for each trial, the probability of winning is the same, and for each trial there are only two possible outcomes, winning and losing.

b) The mean and the standard deviation of X are given as follows:

This means that out of nine friends, the expected number of those who area expected to win a prize is of 1.125, varying around 0.99.

c) At least 3 friends winning a prize is not unusual, as 3 is less than 2.5 standard deviations above the mean.

The binomial distribution represents the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

The change of winning is of one in eight, hence the parameter p is obtained as follows:

p = 1/8 = 0.125.

There are nine friends, hence the parameter n is given as follows:

n = 9.

Then the mean is of:

E(X) = 9 x 0.125 = 1.125.

The standard deviation is of:

[tex]\sqrt{V(X)} = \sqrt{9 \times 0.125 \times 0.875} = 0.99[/tex]

A measure is considered unusual if it is more than 2.5 standard deviations above the mean, hence the upper bound is given as follows:

More than 1.125 + 2.5 x 0.99 = 3.6 friends.

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The simplified form of the given expression is 3(i-2)

A term can be a number, a variable, product of two or more variables or product of a number and a variable. An algebraic expression is formed by a single term or by a group of terms. For example, in the expression 4x + y, the two terms are 4x and y.

Given is an expression, (−8+i)+(2+2i)

Simplifying the expression,

(−8+i)+(2+2i)

= -8 + i + 2 + 2i

= -8+2+2i+i

= -6+31i

= 3(i-2)

Hence, the simplified expression is 3(i-2)

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The equivalent equations are 2 + x = 5,  x + 1 = 4 , Negative 5 + x = negative 2. Option A, B and E

An algebraic expression can be defined as a mathematical expression made up of terms, coefficients, constants, factors and variables.

Algebraic expression are also known to consist arithmetic operations, which includes;

From the information given, we have the equations;

1.  2 + x = 5

collect like terms

x = 3

2. x + 1 = 4

collect like terms

x = 3

3. 9 + x = 6

collect like terms

x = 15

4.  x + (negative 4) = 7

x - 4 = 7

collect like terms

x = 11

5. Negative 5 + x = negative 2

-5 + x = -2

collect like terms

x = 3

Hence, the options are A, B, E

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In actuality, modern electronic calculators with advanced features are specialised computers with a specific function.

The two numbers can be rounded to the closest ten to get an estimate for the calculation 348 + 598.

Rounding 348 to the nearest 10, we get 350.

600 is the result when 598 is rounded up to ten.

Consequently, 350 + 600 = 950 is what we estimate the sum of 348 and 598 to be.

The correct calculation is 348 + 598 = 946 when using a calculator.

In comparison to the actual answer of 946, our guess of 950 is 4 higher. This indicates that we slightly overestimated the result in our estimation. Since the figures are rounded to the closest 10, there isn't much of a change, and this is reasonable.

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The x-intercept of the equation y=-13x+234 is (18,0).

The point slope form is used to find the equation of the straight line which is inclined at a given angle to the x-axis and passes through a given point.

The equation of the point slope form is: (y - y₁) = m(x - x₁)

where, (x, y) is a random point on the line and m is the slope of the line.

1) From the give table, we have (18, 0) and (15, 39).

Here, slope(m) = (39-0)/(15-18)

m= -13

2) Substitute m= -13 and (x₁, y₁)=(18, 0) in point slope form, we get

(y-0)= -13(x-18)

y=-13x+234

To find the x-intercept, substitute in 0 for y and solve for x. To find the y-intercept, substitute in 0 for x and solve for y.

x-intercept(s): (18,0)

y-intercept(s): (0,234)

Therefore, the x-intercept of the equation y=-13x+234 is (18,0).

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To graph the system of equations 3x - 2y = 4 and 3x + 2y = 8, we first need to solve for the value of x and y.

To solve for x and y, we can use the first equation 3x - 2y = 4,

3x - 2y = 4

y = (3x/2) - 2

we can substitute this value of y into the second equation 3x + 2y = 8

3x + 2(3x/2 - 2) = 8

3x + 3x - 4 = 8

6x = 12

x = 2

Now we can substitute the value of x into the first equation to find the value of y

3(2) - 2y = 4

6 - 2y = 4

-2y = -2

y = 1

so the point of intersection is (2,1)

In order to graph the system of equations, we can plot the point of intersection (2,1) and then use the slope-intercept form of the equations to graph the lines.

The slope-intercept form of the first equation is y = (3/2)x - 2

The slope-intercept form of the second equation is y = (1/2)x + 4

So, the lines have a slope of 3/2 and 1/2 respectively.

We can now plot these lines on the coordinate plane, with the point of intersection (2,1) being the point where the lines cross.

The graph of the system of equations is the two lines crossing at the point (2,1)

Answer:

Pythagorean theorem:

[tex]a ^{2} + b ^{2} = c^{2} [/tex]

[tex]2 ^{2} + 11 ^{2} = c^{2} [/tex]

2×2=4 11×11=121

4+121=125

[tex]c = \sqrt{125} [/tex]

Answer:

Step-by-step explanation:

Solving by factoring is a strategy used to solve quadratic equations, which are polynomials of the form ax^2 + bx + c = 0. The goal of this strategy is to factor the quadratic expression on one side of the equation and set each factor equal to zero to find the solutions of the equation.For example, consider the equation x^2 - 6x + 8 = 0. To solve this equation by factoring, we can first find the factors of the constant term (8) which are 1 and 8. And the factors of the leading coefficient (1) and the constant term (8) which are (1,-8) and (1,8). We can then use the distributive property to rewrite the equation as (x - 4)(x - 2) = 0.Once we have factored the quadratic expression, we set each factor equal to zero and solve for x. In this case, x - 4 = 0 and x - 2 = 0. Solving for x, we get x = 4 and x = 2. These are the solutions to the equation.We can check our solutions by plugging them back into the original equation to see if they make the equation true.x = 4 and x = 2 are the solutions of the equation x^2 - 6x + 8 = 0.It's important to note that factoring can also be used to solve equations that are not in the standard form, like x^2 - 6x + 8 = 12 by subtracting 12 from both sides of the equation, and then factor the left side.In summary, factoring is a strategy used to solve quadratic equations by factoring the quadratic expression on one side of the equation, setting each factor equal to zero, and solving for x. The solutions of the equation are the values of x that make the equation true.

Using substitution method, the values of a and b in the system of linear equations are 6.3 and 3.65

A system of linear equations is a set of two or more linear equations that contain two or more variables. The goal of solving a system of linear equations is to find the values of the variables that satisfy all of the equations in the system simultaneously. Linear equations are equations that can be written in the form of ax + b = y, where a, b, and x are constants and x is the variable.

To solve this problem, we have to solve the equations simultaneously using either substitution or elimination method.

0.95a + 1.1b = 10 ...eq(i)

a + b = 9.95 ...eq(ii)

From equation (ii)

a + b = 9.95

a = 9.95 - b ...eq(iii)

Put eq(iii) into eq(i)

0.95(9.95 - b) + 1.1b = 10

b = 3.65

put b = 3.65 into eq(ii)

a + 3.65 = 9.95

a = 6.3

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Beginning at the y-intercept (0,35),Line is solid, slopes down one and over one, and is shaded beneath.

Choose your variables first.

If x is the amount of hours spent mowing lawns,

Let y equal the number of hours worked at the clothes business.

A minimum of $400 must be earned from both employment (400 or more)

15x + 10y ≥ 400

In order to graph y = -15/10x +35, get y.

10 hours minimum every day in the store.

35 hours maximum per week

x + y ≤ 35

To graph y = -x + 35, find y.

Making the graph: 0 to 35 for the x and y axes.

First equation: y = -15/10x + 35 y-intercept = 35 slope: solid line on the y-axis starting at 35, going down 12, over 10 Shade above solid line

Second formula: y 10

At y=10, draw a solid horizontal line; more than should be shaded above the line.

Third formula: y = -x + 35

beginning at the y-intercept (0,35)

Line is solid, slopes down one and over one, and is shaded beneath.

The overlapping region on the graph will appear as a little triangle with its points under the line from the third equation, above the horizontal line, and above the line from the first equation.

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Answer:

A: 12.5

B: 36

C: 40

D: 88

Step-by-step explanation:

those are the answers

The mapping diagram above represents a function since for each number in Set A(input), there is a single mapping in Set B(output).

To verify whether a relation represents a function, we need to observe if each input is mapped to only one output.

The input and output sets for this problem are given as follows:

Then the mappings are given as follows:

As each input is mapped to a single output, the relation represents a function, and the blanks are completed considering this.

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The value of {d} for F = 4 is equivalent to {d} = 7.9.

Given is that {F} is inversely proportional to {d} to the power of 2.

We can write the inverse relation as -

F [tex]$\alpha[/tex] 1/d²

Rewriting the relation replacing constant, we get -

F = K/d²

For {d} = 6, {F} = 7. We can write -

F = K/d²

7 = K/36

K = 36 x 7

K = 252

Now, we get the relation as -

F = 252/d²

For {F = 4}, we can write -

4 = 252/d²

d² = 252/4

d² = 63

d = 7.9

Therefore, the value of {d} for F = 4 is equivalent to {d} = 7.9.

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Answer:

nice

Step-by-step explanation:

The value of the expression P(x) in the function for the weight of a thin water layer between heights x and Δx, P(x)·Δx, is;  P(x) = 47040·x N/m

A function is a rule that defines the relationship between a set of input and outputs variables, such that each input variable is mapped to only one output variable.

The parameters of the right prism are presented as follows;

Height of the isosceles triangular face = 5 meters

Width of the triangle = 4 meters

The length of the right prism = 6 meters

The function for the weight of a thin water layer between height x and height x + Δx = P(x)·Δx

Weight = Density × Volume × Gravity

Volume = Cross sectional area × Height

Cross sectional area = Width × Length = Width × 6

Using similar triangles, we get;

Width, W = (4/5)×x

Therefore; Cross sectional area = (4/5)×x × 6

Weight = Density × Cross sectional area × Gravity × Height

The height of the section = Δx, therefore;

Weight = Density × Cross sectional area × Gravity × Δx = P(x) × Δx

Therefore; Density × Cross sectional area × Gravity = P(x)

P(x) = ρ × (4/5) × x × 6 × g

The density of water, ρ = 1000 kg/m³

g = 9.8 m/s²

P(x) = 1000 kg/m³ × (4/5) × x × 6 m × 9.8 m/s² = 47040·x N/m

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Answer:

10 cupcakes

Step-by-step explanation:

6 cupcakes have mint icing. This means there are 24 cupcakes left. 7/12 of these, or 14 cupcakes, have chocolate frosting. This means that there are 10 vanilla frosted cupcakes.

The following are the values for the length of each rectangles:

(a) L = 37.5 m

(b) L = 26.2 cm

(c) L = 25.5 km

(d) L = 20 m

In calculating for the area of rectangle, the breadth is multiplied by the length of the rectangle.

We shall evaluate for the Lengths of the rectangles as follows:

Length = 375 m²/ 10 m

Length = 37.5 m

Length = 393 cm²/ 15 cm

Length = 26.2 cm

Length = 280.5 km²/ 11 km

Length = 25.5 km

Length = 340 m²/ 17 m

Length = 20 m

Hence, 37.5 m, 26.2 cm, 25.5 km, and 20 m are the respective values for the length of rectangles with area 375 m², 393 cm², 280.5 km², and 340 m².

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Using circle theorems, the value of x in the cyclic quadrilateral is 6 units

A cyclic quadrilateral is a four-sided shape whose vertices all lie on the same circle. This means that all four angles of the quadrilateral are equal to one another.

This is further explained by circle theorem which is a branch of mathematics that deals with the properties of circles and their relationships with other shapes. It is based on the fact that all points on a circle are equidistant from the center of the circle. Circle theorem has many applications, such as finding the area of a circle, calculating the circumference of a circle, and even determining the angles of a triangle formed by three points on a circle.

In this given problem, to find the value of x, we have to use proportions or ratios of the sides given.

This becomes;

(x + 6) / x = 12 / x

when the denominators cancel out;

x + 6 = 12

Collect like terms

x = 12 - 6

x = 6

The value of x is 6

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Answer: y = -3x + 2

Step-by-step explanation: The slope-intercept form of the linear equation is y = mx + b, where m is the slope and b is the y-intercept.

To find the slope (m) of the linear function, we can use the formula:

m = (change in y) / (change in x)

By using two points in the table (-3,11) and (2,-4), we can find the slope:

m = ( -4 - 11) / ( 2 - (-3)) = -15 / 5 = -3

Now, we know the slope, we can use any point from the table to find the y-intercept (b) by substituting the point's x and y values into the equation and solving for b:

y = -3x + b

using the point (-3, 11)

11 = -3 * (-3) + b

11 = 9 + b

b = 2

So the equation of the linear function represented by the table in slope-intercept form is:

y = -3x + 2

The value of the tangent of angle B where the tangent of angle A is 1/2 and ∠A and ∠B are the non-right angles in the right triangle is; tan B = 2

A right triangle is a triangle that has the measure of one of the interior angles as 90°.

The angles in the right triangle that are not right angles = A and B

tan(A) = 1/2

The value tan B

Angle ∠A and angle ∠B are complementary angles

Therefore; m∠A + m∠B = 90°

The tangent trigonometric ratio indicates that the tangent of a non-right angle in a right triangle is; (The length of the opposite side)/(The length of the adjacent side)

Let a represent the length of the side opposite to angle ∠A, and let b represent the length of the side opposite to the angle ∠B, we get;

tan(A) = a/b

tan (B) = b/a

Therefore, if tan(A) = 1/2, tan(B) = The inverse of tan(A) = 1/(1/2) = 2

tan B = 2

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Answer:

14

Step-by-step explanation:

Basically, x is substituted by 10.

So if you plug that into the function, you would get

[tex]f(10)=2(10)-6[/tex]

Simplify it and you get

[tex]f(10)=14[/tex]

Answer: 150 percent.

Step-by-step explanation:

The problem asks for the percent increase in the annual student fees from 2000 to 2014.

First, we need to know how to find a percent increase.

The percentage change formula is

(Increase in amount) / (Original amount) * 100

Using this formula, we can solve this problem.

Since the fee in 2000 was $6000 and the fee in 2014 was $15000, we can find the increase in fees.

Increase in fees = 15000-6000 == 9000

Next, let's use the formula to find the answer.

Percentage Increase = (Increase in fees) / (Original amount) * 100

= 9000 / 6000 * 100

= 150

Therefore, the answer is 150 percent.

The required solution of the given equation would be w = (x - 2y)/2 in terms of x and y.

What is an equation?

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

The equation is given in the question, as follows:

2y + 2w = x

Solving the above equation in terms of x and y

As per the question, we have

2y + 2w = x

Subtract 2y both sides of the equation, and we get

2w = x - 2y

w = (x - 2y)/2

Thus, the required equation would be w = (x - 2y)/2.

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Answer:

First question

AC, AB,CB,CD,BD

Second question

KM, KL,ML,LN,MN

Step-by-step explanation:

Observe the different lengths of the triangles and where they connect the longest side of the little triangle is the shortest side of the biggest one

The arithmetic sequence's fifth term is therefore 84.

An arithmetic sequence's nth term is determined by a = a + (n - 1)d. So, enter the values a = 2 and d = 3 into the formula to determine the nth term.

The formula to determine the nth term of an arithmetic series is a = a1 + (n - 1)d if a1 is the first term and d is the common difference.

In the mathematical sequence, 84 is the fifth term.

The process for calculating value

An arithmetic sequence's nth term can be calculated using the following formula:

Tn = a + (n -1)d

Where;

First term is a.

N words are present.

d is the typical difference

We must substitute nas 5's value in order to find the fifth term.

T5 = 4 + (5-1) 20

T5 = 4 + 4(20) (20)

the bracket be expanded

T5 = 4 + 80

T5 = 84

The arithmetic sequence's fifth term is therefore 84.

The complete question is,

4 is the initial term and 20 is the constant difference in an arithmetic series. Discover the sequence's sixth phrase.

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Answer:

x < -16.714... or x > 2.428...

Step-by-step explanation:

|0.7x + 5| > 6.7

=> 0.7x + 5 = -6.7 or 0.7x + 5 = 6.7

=> 0.7x = -6.7 - 5 or 0.7x = 6.7 - 5

=> x = -11.7/0.7 or x = 1.7/0.7

=> x = -16.714... or x = 2.428...

note: if |u| > a, a > 0 then u < -a or u > a

Answer:

Not sure what you meant and could of used some specifics but I got this with my experience in algebra and geometry.

9906513923.77

98765 times triangle A 32598765÷325

Answer:

A

Step-by-step explanation:

-3 x 4 = -12

[tex]10^{3}[/tex] x [tex]10^{6}[/tex] = [tex]10^{9}[/tex]  You add the exponents when you are multiplying powers with the same bases.